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Properties

Label 187.2.p.a
Level $187$
Weight $2$
Character orbit 187.p
Analytic conductor $1.493$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [187,2,Mod(4,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([4, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("187.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 187.p (of order \(20\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.49320251780\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6} - 24 q^{10} - 40 q^{13} + 2 q^{14} - 48 q^{16} - 18 q^{17} - 2 q^{20} + 16 q^{21} - 70 q^{22} - 16 q^{23} + 28 q^{24} - 22 q^{27} + 42 q^{28} - 2 q^{29} - 44 q^{30} - 6 q^{31} + 32 q^{33} + 44 q^{34} + 12 q^{35} + 30 q^{37} - 80 q^{38} + 78 q^{39} - 100 q^{40} - 56 q^{41} + 52 q^{44} - 68 q^{45} + 14 q^{46} - 16 q^{47} - 110 q^{48} + 84 q^{50} + 14 q^{51} - 100 q^{52} - 20 q^{54} - 84 q^{55} + 36 q^{56} - 48 q^{57} - 26 q^{58} + 28 q^{61} + 108 q^{62} - 40 q^{63} + 120 q^{64} + 28 q^{65} - 48 q^{67} + 102 q^{68} + 24 q^{69} + 2 q^{71} + 80 q^{72} - 30 q^{73} - 28 q^{74} - 80 q^{75} - 104 q^{78} + 44 q^{79} - 92 q^{80} + 140 q^{81} - 28 q^{82} - 52 q^{84} + 76 q^{85} + 12 q^{86} + 50 q^{88} - 32 q^{89} + 204 q^{90} + 42 q^{91} + 2 q^{92} + 16 q^{95} + 240 q^{96} - 34 q^{97} + 24 q^{98} - 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.64595 + 2.26545i 0.433352 + 0.850501i −1.80509 − 5.55550i 0.526458 − 3.32392i −2.64004 − 0.418141i −0.503952 + 0.989061i 10.2304 + 3.32405i 1.22780 − 1.68992i 6.66366 + 6.66366i
4.2 −1.38201 + 1.90217i 1.08670 + 2.13277i −1.09027 − 3.35550i −0.583785 + 3.68587i −5.55871 − 0.880413i 0.909545 − 1.78508i 3.41722 + 1.11032i −1.60442 + 2.20830i −6.20435 − 6.20435i
4.3 −1.31417 + 1.80880i −0.639016 − 1.25414i −0.926682 − 2.85203i −0.322855 + 2.03843i 3.10826 + 0.492300i −1.13928 + 2.23597i 2.12384 + 0.690076i 0.598833 − 0.824223i −3.26282 − 3.26282i
4.4 −1.04806 + 1.44253i −0.804161 − 1.57826i −0.364434 − 1.12161i −0.116829 + 0.737628i 3.11950 + 0.494080i 1.20459 − 2.36415i −1.39168 − 0.452186i −0.0808598 + 0.111294i −0.941609 − 0.941609i
4.5 −0.887944 + 1.22215i −0.0267877 − 0.0525737i −0.0871729 − 0.268291i 0.507206 − 3.20238i 0.0880390 + 0.0139440i 1.63147 − 3.20194i −2.46815 − 0.801951i 1.76131 − 2.42423i 3.46341 + 3.46341i
4.6 −0.830083 + 1.14251i 0.831391 + 1.63170i 0.00173994 + 0.00535500i 0.0745546 − 0.470719i −2.55435 − 0.404570i −1.69465 + 3.32594i −2.69377 − 0.875258i −0.207866 + 0.286102i 0.475915 + 0.475915i
4.7 −0.273504 + 0.376447i 0.441712 + 0.866909i 0.551127 + 1.69619i −0.114620 + 0.723685i −0.447155 − 0.0708224i −0.487084 + 0.955955i −1.67434 − 0.544026i 1.20693 − 1.66120i −0.241080 − 0.241080i
4.8 −0.212815 + 0.292914i −1.13391 − 2.22542i 0.577525 + 1.77744i −0.417832 + 2.63808i 0.893168 + 0.141464i −1.09590 + 2.15082i −1.33223 − 0.432866i −1.90338 + 2.61978i −0.683811 − 0.683811i
4.9 −0.0756214 + 0.104084i 1.34406 + 2.63788i 0.612919 + 1.88637i 0.0370110 − 0.233679i −0.376200 − 0.0595843i 2.00593 − 3.93685i −0.487406 − 0.158368i −3.38852 + 4.66390i 0.0215233 + 0.0215233i
4.10 0.201795 − 0.277747i −0.935793 − 1.83660i 0.581612 + 1.79002i 0.314547 − 1.98597i −0.698947 − 0.110702i 0.501695 − 0.984632i 1.26756 + 0.411855i −0.734024 + 1.01030i −0.488124 − 0.488124i
4.11 0.668874 − 0.920626i −0.202818 − 0.398052i 0.217874 + 0.670546i −0.507205 + 3.20236i −0.502117 − 0.0795275i 2.13204 − 4.18437i 2.92758 + 0.951227i 1.64605 − 2.26559i 2.60893 + 2.60893i
4.12 0.765751 − 1.05397i 1.05793 + 2.07631i 0.0935641 + 0.287961i 0.683608 − 4.31613i 2.99848 + 0.474912i −1.20165 + 2.35837i 2.85317 + 0.927052i −1.42849 + 1.96614i −4.02558 − 4.02558i
4.13 0.827467 − 1.13891i 0.330414 + 0.648475i 0.00561864 + 0.0172924i −0.427401 + 2.69851i 1.01196 + 0.160279i −1.38515 + 2.71850i 2.70208 + 0.877960i 1.45201 − 1.99852i 2.71970 + 2.71970i
4.14 1.09096 − 1.50157i −1.13445 − 2.22649i −0.446504 − 1.37420i 0.176770 − 1.11608i −4.58088 − 0.725540i −0.144965 + 0.284510i 0.979837 + 0.318368i −1.90691 + 2.62463i −1.48304 − 1.48304i
4.15 1.44335 − 1.98660i −0.272661 − 0.535128i −1.24529 − 3.83260i 0.0688455 − 0.434673i −1.45663 − 0.230707i −1.33328 + 2.61671i −4.74045 − 1.54026i 1.55134 − 2.13523i −0.764154 − 0.764154i
4.16 1.49639 − 2.05960i 1.05967 + 2.07972i −1.38474 − 4.26180i −0.119706 + 0.755793i 5.86908 + 0.929571i 0.600644 − 1.17883i −6.00731 − 1.95189i −1.43899 + 1.98060i 1.37751 + 1.37751i
38.1 −2.30228 + 0.748055i 1.85991 − 0.294580i 3.12286 − 2.26889i 1.21515 + 2.38487i −4.06166 + 2.06952i 0.345011 + 0.0546444i −2.64667 + 3.64282i 0.519302 − 0.168732i −4.58163 − 4.58163i
38.2 −1.96554 + 0.638643i −1.63407 + 0.258811i 1.83745 − 1.33498i 0.781962 + 1.53469i 3.04654 − 1.55229i 3.21586 + 0.509342i −0.329459 + 0.453461i −0.249973 + 0.0812212i −2.51709 − 2.51709i
38.3 −1.93141 + 0.627553i −2.69128 + 0.426258i 1.71848 − 1.24855i −0.435016 − 0.853767i 4.93047 − 2.51220i −4.37611 − 0.693108i −0.148208 + 0.203990i 4.20815 − 1.36731i 1.37598 + 1.37598i
38.4 −1.61464 + 0.524629i 0.410158 − 0.0649626i 0.713800 − 0.518606i −0.824518 − 1.61821i −0.628176 + 0.320072i −0.362922 − 0.0574812i 1.11535 − 1.53515i −2.68916 + 0.873761i 2.18026 + 2.18026i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.c even 4 1 inner
187.p even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 187.2.p.a ✓ 128
11.c even 5 1 inner 187.2.p.a ✓ 128
17.c even 4 1 inner 187.2.p.a ✓ 128
187.p even 20 1 inner 187.2.p.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.2.p.a ✓ 128 1.a even 1 1 trivial
187.2.p.a ✓ 128 11.c even 5 1 inner
187.2.p.a ✓ 128 17.c even 4 1 inner
187.2.p.a ✓ 128 187.p even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(187, [\chi])\).